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Yes, you're right. As for why the harmonic series doesn't produce notes that work in all keys, the simple answer is that the math just doesn't add up.
Let's work out the math for just intonation: Suppose you choose X Hz for the fundamental frequency and go from there. Then the octave above the fundamental should have frequency 2 X Hz. Meanwhile, the ...

Essentially all instruments produce overtones, which are frequencies other than the dominant frequency of the note. When one or more overtones is a multiple of the base (or fundamental) frequency, it's called a harmonic. Some instruments like drums do not generally have harmonic overtones. Others like guitars, violins, and flutes do have have them; more ...

It is not true in general that the higher you go on the fret board, the lower your harmonic is. Actually, if your were to play an harmonic at the 24th fret, you would hear a note sounding an octave higher than the harmonic at the 12th.
Still, however, the harmonics behave differently than fretted notes. Now, let’s get physical and explain why. On perfect ...

It's a simple question with quite an involved answer.
First a quick primer on wavelength, frequency and pitch.
Frequency is how many times a vibration repeats in a period of time. Wavelength is the length of a single vibration, which for something like a guitar string, corresponds to the length of time it takes to repeat. You can see that if the ...

Alex Basson has given you a great introduction to the mathematics. Let me approach the answer from a different perspective, that of the performing musician in a historical context.
Setting the mathematics aside, to put it simply, just intonation is what happens when you have a group of singers performing a capella, or a string quartet, or any other ensemble ...

(Just in case of any confusion, this answer was originally in response to a duplicate question about pinch harmonics used by Billy Gibbons of ZZTop - thanks for the merge, Dr Mayhem!)
These are probably pinch harmonics. In which case, they are not produced by a particular amp or effects setting, but are produced by the picking hand while playing (although ...

Harmonics occur at the whole-number ratios of the length of the string, e.g. 1/2 the length of the string, 1/3 the length of the string, 2/3 the length of the string, 1/4, 3/4, etc. The lower the numbers involved, the easier the harmonic will be to play. So for example, theoretically there's a harmonic at 6/19 the length of the string, but it will be much ...

You might be actually playing A 110, two octaves below A 440.
The open A string on a standard tuned guitar is actually two octaves below the A that is normally tuned to 440. To play the A that should be at 440 Hz, you have to play the 5th fret on the high E string, or the 10th on the B string, or the 14th on the G string, etc.
Why? The 440 A is the A above ...

There is indeed a reason! The notes you play on a trumpet with a particular fingering come from the harmonic series, which is a series of tones based on the root, or fundamental frequency.
The idea is that the harmonics (also called overtones) are whole-number multiples of the fundamental frequency. If the fundamental frequency of, say, your trumpet, is ...

WHY do harmonics happen?
Well, they don't happen – not necessarily.
Strings can do all kind of stuff:
All these are possible ways a snapshot of a moving string could look like. Not only that, it could also move at each point with an arbitrary velocity, including the possibility of a velocity that's everywhere 0 (for an infinitesimally short ...

You can't actually do this - obviously you are aware that the whole pich/natural/articial harmonic bit doesn't create frequencies, it just cuts out some so the tonal quality changes.
You do have technological alternatives though, if you do want to build lower frequencies related to the original root.
Octave dividers are the simplest example - you can buy ...

A concise way to put it is:
The vibrating part of the string vibrates to accomodate for all nodes you apply to it, while keeping a constant distance between all nodes. Additionally, the string vibrates with the least number of nodes possible, since this is what takes the least energy.
When you play an open string, the constraints on the string are only ...

Consonants are primarily just noise—largely high, semi-random frequencies. Vowels are generally the only part of speech that have harmonic content as opposed to the inharmonic content of consonants. Bright, forward vowels like EEEEEE have way more overtones than dark, back vowels like AHHHHHH. Most consonants other than the nasals are so brief that I'd ...

You need to change your finger position on the pick so that your index finger or thumb is slightly over hanging the side of the pick. Strike the note with the pick and then brush with the flesh of your finger or thumb (depending on whether it's an up or down stroke). Notes / strings will get different harmonics at slightly different places. Use your pick ups ...

There should be a (slightly flat) C as the 7th harmonic on the D string. You should find it at all six places where you would cut the string if you were to cut it into seven equal lenght pieces: for instance between the second and third fret, about one third from the third fret.
However it might be hard to nail with a decent sound, and you will most likely ...

Use flats. In fact, write the bass of the 3rd chord as D♭. You're not only running parallel second inversion diminished chords downwards, you're doing so in a key that already uses flats in the key signature. I'd even be tempted to write the soprano and bass of the second chord as C♭ and E♭♭ respectively, but leaving them be might be a little easier to read. ...

As leftaroundabout indicated in another answer, you could achieve subharmonics (specifically a difference tone) using two sound sources - such as two strings.
See "Combination tone" on wikipedia for further specifics.
Or see "Resultant tone" regarding the usage of this for organs, where sometimes two shorter organ pipes are used to sound simultaneously ...

Technically speaking, the answer is infinity for all intervals.
This is because for any resonant harmonic of a fundamental, a harmonic exists at twice the frequency.
There is an order in which these intervals appear, and that is easily found by looking at the harmonic series. You do need to know, of course, that the 12-tone equal temperament that we use ...

We first need to categorize each interval, assign it a "consonance amount". That's the first problem we find. In the case of the fourth, for example, some consider it perfect consonance, and others consider it a dissonance, depending context (and who you ask).
For simplicity, let's define ours based on Wikipedia's:
1: Perfect consonances: unison, octave, ...

In the context of acoustic analysis:
Overtone: any resonant frequency above the fundamental frequency.
Harmonic: resonant frequency that is an integer multiple of the fundamental frequency.
For almost any^* musical instrument, any time you play a sound, you get a whole series of overtones. The feature of pitched instruments that makes them pitched is ...

I have come across an even more astonishing system for producing pure intervals on a guitar. A Turkish guitarist, Tolgahan Çoğulu, has patented a system for building a guitar that has channels under each string position that allows the quick installation or removal of any number of tiny partial frets, each one string-space wide, which can be adjusted up or ...

It appears to be the 'fret' number, just as you'd find in the rest of the tab. Except open harmonics start out over frets - 1st is at the 12th fret, an octave, but as the string gets divided more and more, the nodes become out of line with the guitar frets. The 6th harmonic is found over 'fret 2.7' and the 5th over 'fret 3.2'.This is obviously very close to ...

The difference is quite simple, and we might be over-complicating it in other answers.
Overtone: any resonant frequency above the fundamental frequency.
Harmonic: resonant frequency that is an integer multiple of the fundamental frequency.
A harmonic is a type of overtone. All resonant frequencies above the fundamental are overtones, but only the ones ...

We hear harmonics because they are physically produced by the instrument; they are not "invented" as some sort of illusion. In fact, we often aren't consciously aware of them, though we can hear their effect on an instrument's timbre, or tone quality.
They are caused because when an instrument such a string vibrates, it actually does so at more than one ...

The spectral effect of hard sync is incredibly varied and not as systematic as AM or FM. However, it is definitely capable of producing inharmonic spectra. Here's an excellent article explaining how hard sync can make synthesis of an acoustic piano more realistic by creating inharmonic sounds reminiscent of the striking of the string by the hammer and such: ...

In my answer to How do harmonics work?, I said:
[A] vibrating guitar string has components at many multiples of the base frequency (call it F). To your ear it still sounds like the fundamental, but mathematically it's more like this:
a*F + b*2F + c*3F + ...
The higher-frequency elements give the note it's timbre; this is how you can ...

Actually, the opposite question is more relevant
why doesn't a string vibrate at all the frequencies? When you hit it, you aren't providing just one vibration node, but putting in a chunk of energy.
It comes down to where the nodes, or zeros are (have a look at @MatthewRead's answer over here) as with the ends of the string fixed, any wave which has an ...

Resonant Frequency
Good guitars have the wooden top and body of the guitar carefully carved to resonate at a certain single frequency.
While the luthier is carving and bracing the top of the guitar, long before the guitar is assembled and the strings are put on, he repeatedly taps the top with a knuckle, as if it were a drum, and listens to the fundamental ...

Harmonics occur on the string at points that divide the string into equal parts. Lightly touch a string right above the 12th fret bar and while doing so strum the string -- you'll notice that the string sounds an octave higher, though different from just fretting the 12th. What you're doing is dampening the fundamental frequency and "odd" harmonics (3x, ...